What do neural connection weights represent 'conceptually'?
I don't know if my intuition is correct but I will give it a try. 

You could see weights as how much important one thing is, the problem is to understand what that thing represents. When I say thing I'm referring to the output of a specific neuron. I don't think that we can say what the output of a neuron represents in the real world unless we directly relate it through an error function or if the function used to compute that particular value have some meaning in the real world.

Edit:

If you want, you could actually build your neural network such that its neurons represent something. It's also very simple. you have only to write down all the equations relative to that particular topic. You could put them in a big system or, and this is better, you could put them in several systems such that the outputs of system 1 are the input of system 2 and so on. You could convert each system into a layer where each neuron represents an equation. Note that in this case, you would have the classical neuron with


z = dot(w.T,x) + b
a = g(x)


but a more complex equation for z (but still based on weights) and a linear activation function for a. In this case, you could name each neuron and say what they represent in the real world. 

However, this isn't the purpose of a neural network. A neural network should have neurons with simple equations to be fast thus the linear interpolating function dot(w.T,x) + b is the best choice (the fact that the activation function is almost always non linear and in some cases a non-banal function is due to other thing and could be an interesting question). A neural network should also be as general as possible because usually is build upon a system that you don't know completely.

So I modify slightly my answer: is not simply that you don't know what a neuron represent, excluding the ones of the output layer, you don't want that they have a meaning in the real world.